1. Is there analog of Lyndon–Hochschild–Serre spectral sequence for not normal subgroup?
2. What can you say about it? Can you describe $E^{p, q}_1$ ? What is about $E^{p, q}_2$?
3. What is the best technique to get the spectral sequence? For me Grothendieck spectral sequence much better than spectral sequence of a filtered complex.

There is a parallel question which is likely easier.

1. Is there analog of Hochschild–Serre spectral sequence for Lie subalgebra which is not ideal?

2 and 3 remain the same

I already asked a version of this question but get no responses.

http://math.stackexchange.com/questions/1112179/hochschild-serre-spectral-sequence-for-not-normal-subalgebra